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## G = C56⋊(C2×C4)  order 448 = 26·7

### 10th semidirect product of C56 and C2×C4 acting via C2×C4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — C56⋊(C2×C4)
 Chief series C1 — C7 — C14 — C2×C14 — C2×C28 — C2×C4×D7 — C2×C8⋊D7 — C56⋊(C2×C4)
 Lower central C7 — C14 — C28 — C56⋊(C2×C4)
 Upper central C1 — C22 — C2×C4 — C2.D8

Generators and relations for C56⋊(C2×C4)
G = < a,b,c | a56=b2=c4=1, bab=a13, cac-1=a15, bc=cb >

Subgroups: 556 in 118 conjugacy classes, 55 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, C23, D7, C14, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C4.Q8, C2.D8, C2.D8, C2×C4⋊C4, C42⋊C2, C2×M4(2), C7⋊C8, C56, C4×D7, C4×D7, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×D7, M4(2)⋊C4, C8⋊D7, C2×C7⋊C8, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C7×C4⋊C4, C2×C56, C2×C4×D7, C2×C4×D7, C28.Q8, C4.Dic14, C8⋊Dic7, C7×C2.D8, D7×C4⋊C4, C4⋊C47D7, C2×C8⋊D7, C56⋊(C2×C4)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D7, C4⋊C4, C22×C4, C2×D4, C2×Q8, D14, C2×C4⋊C4, C8⋊C22, C8.C22, C4×D7, C22×D7, M4(2)⋊C4, C2×C4×D7, D4×D7, Q8×D7, D7×C4⋊C4, D8⋊D7, Q16⋊D7, C56⋊(C2×C4)

Smallest permutation representation of C56⋊(C2×C4)
On 224 points
Generators in S224
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168)(169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224)
(2 14)(3 27)(4 40)(5 53)(6 10)(7 23)(8 36)(9 49)(11 19)(12 32)(13 45)(16 28)(17 41)(18 54)(20 24)(21 37)(22 50)(25 33)(26 46)(30 42)(31 55)(34 38)(35 51)(39 47)(44 56)(48 52)(57 77)(58 90)(59 103)(61 73)(62 86)(63 99)(64 112)(65 69)(66 82)(67 95)(68 108)(70 78)(71 91)(72 104)(75 87)(76 100)(79 83)(80 96)(81 109)(84 92)(85 105)(89 101)(93 97)(94 110)(98 106)(107 111)(113 129)(114 142)(115 155)(116 168)(117 125)(118 138)(119 151)(120 164)(122 134)(123 147)(124 160)(126 130)(127 143)(128 156)(131 139)(132 152)(133 165)(136 148)(137 161)(140 144)(141 157)(145 153)(146 166)(150 162)(154 158)(159 167)(169 177)(170 190)(171 203)(172 216)(174 186)(175 199)(176 212)(178 182)(179 195)(180 208)(181 221)(183 191)(184 204)(185 217)(188 200)(189 213)(192 196)(193 209)(194 222)(197 205)(198 218)(202 214)(206 210)(207 223)(211 219)(220 224)
(1 201 60 149)(2 216 61 164)(3 175 62 123)(4 190 63 138)(5 205 64 153)(6 220 65 168)(7 179 66 127)(8 194 67 142)(9 209 68 157)(10 224 69 116)(11 183 70 131)(12 198 71 146)(13 213 72 161)(14 172 73 120)(15 187 74 135)(16 202 75 150)(17 217 76 165)(18 176 77 124)(19 191 78 139)(20 206 79 154)(21 221 80 113)(22 180 81 128)(23 195 82 143)(24 210 83 158)(25 169 84 117)(26 184 85 132)(27 199 86 147)(28 214 87 162)(29 173 88 121)(30 188 89 136)(31 203 90 151)(32 218 91 166)(33 177 92 125)(34 192 93 140)(35 207 94 155)(36 222 95 114)(37 181 96 129)(38 196 97 144)(39 211 98 159)(40 170 99 118)(41 185 100 133)(42 200 101 148)(43 215 102 163)(44 174 103 122)(45 189 104 137)(46 204 105 152)(47 219 106 167)(48 178 107 126)(49 193 108 141)(50 208 109 156)(51 223 110 115)(52 182 111 130)(53 197 112 145)(54 212 57 160)(55 171 58 119)(56 186 59 134)

G:=sub<Sym(224)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224), (2,14)(3,27)(4,40)(5,53)(6,10)(7,23)(8,36)(9,49)(11,19)(12,32)(13,45)(16,28)(17,41)(18,54)(20,24)(21,37)(22,50)(25,33)(26,46)(30,42)(31,55)(34,38)(35,51)(39,47)(44,56)(48,52)(57,77)(58,90)(59,103)(61,73)(62,86)(63,99)(64,112)(65,69)(66,82)(67,95)(68,108)(70,78)(71,91)(72,104)(75,87)(76,100)(79,83)(80,96)(81,109)(84,92)(85,105)(89,101)(93,97)(94,110)(98,106)(107,111)(113,129)(114,142)(115,155)(116,168)(117,125)(118,138)(119,151)(120,164)(122,134)(123,147)(124,160)(126,130)(127,143)(128,156)(131,139)(132,152)(133,165)(136,148)(137,161)(140,144)(141,157)(145,153)(146,166)(150,162)(154,158)(159,167)(169,177)(170,190)(171,203)(172,216)(174,186)(175,199)(176,212)(178,182)(179,195)(180,208)(181,221)(183,191)(184,204)(185,217)(188,200)(189,213)(192,196)(193,209)(194,222)(197,205)(198,218)(202,214)(206,210)(207,223)(211,219)(220,224), (1,201,60,149)(2,216,61,164)(3,175,62,123)(4,190,63,138)(5,205,64,153)(6,220,65,168)(7,179,66,127)(8,194,67,142)(9,209,68,157)(10,224,69,116)(11,183,70,131)(12,198,71,146)(13,213,72,161)(14,172,73,120)(15,187,74,135)(16,202,75,150)(17,217,76,165)(18,176,77,124)(19,191,78,139)(20,206,79,154)(21,221,80,113)(22,180,81,128)(23,195,82,143)(24,210,83,158)(25,169,84,117)(26,184,85,132)(27,199,86,147)(28,214,87,162)(29,173,88,121)(30,188,89,136)(31,203,90,151)(32,218,91,166)(33,177,92,125)(34,192,93,140)(35,207,94,155)(36,222,95,114)(37,181,96,129)(38,196,97,144)(39,211,98,159)(40,170,99,118)(41,185,100,133)(42,200,101,148)(43,215,102,163)(44,174,103,122)(45,189,104,137)(46,204,105,152)(47,219,106,167)(48,178,107,126)(49,193,108,141)(50,208,109,156)(51,223,110,115)(52,182,111,130)(53,197,112,145)(54,212,57,160)(55,171,58,119)(56,186,59,134)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224), (2,14)(3,27)(4,40)(5,53)(6,10)(7,23)(8,36)(9,49)(11,19)(12,32)(13,45)(16,28)(17,41)(18,54)(20,24)(21,37)(22,50)(25,33)(26,46)(30,42)(31,55)(34,38)(35,51)(39,47)(44,56)(48,52)(57,77)(58,90)(59,103)(61,73)(62,86)(63,99)(64,112)(65,69)(66,82)(67,95)(68,108)(70,78)(71,91)(72,104)(75,87)(76,100)(79,83)(80,96)(81,109)(84,92)(85,105)(89,101)(93,97)(94,110)(98,106)(107,111)(113,129)(114,142)(115,155)(116,168)(117,125)(118,138)(119,151)(120,164)(122,134)(123,147)(124,160)(126,130)(127,143)(128,156)(131,139)(132,152)(133,165)(136,148)(137,161)(140,144)(141,157)(145,153)(146,166)(150,162)(154,158)(159,167)(169,177)(170,190)(171,203)(172,216)(174,186)(175,199)(176,212)(178,182)(179,195)(180,208)(181,221)(183,191)(184,204)(185,217)(188,200)(189,213)(192,196)(193,209)(194,222)(197,205)(198,218)(202,214)(206,210)(207,223)(211,219)(220,224), (1,201,60,149)(2,216,61,164)(3,175,62,123)(4,190,63,138)(5,205,64,153)(6,220,65,168)(7,179,66,127)(8,194,67,142)(9,209,68,157)(10,224,69,116)(11,183,70,131)(12,198,71,146)(13,213,72,161)(14,172,73,120)(15,187,74,135)(16,202,75,150)(17,217,76,165)(18,176,77,124)(19,191,78,139)(20,206,79,154)(21,221,80,113)(22,180,81,128)(23,195,82,143)(24,210,83,158)(25,169,84,117)(26,184,85,132)(27,199,86,147)(28,214,87,162)(29,173,88,121)(30,188,89,136)(31,203,90,151)(32,218,91,166)(33,177,92,125)(34,192,93,140)(35,207,94,155)(36,222,95,114)(37,181,96,129)(38,196,97,144)(39,211,98,159)(40,170,99,118)(41,185,100,133)(42,200,101,148)(43,215,102,163)(44,174,103,122)(45,189,104,137)(46,204,105,152)(47,219,106,167)(48,178,107,126)(49,193,108,141)(50,208,109,156)(51,223,110,115)(52,182,111,130)(53,197,112,145)(54,212,57,160)(55,171,58,119)(56,186,59,134) );

G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168),(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224)], [(2,14),(3,27),(4,40),(5,53),(6,10),(7,23),(8,36),(9,49),(11,19),(12,32),(13,45),(16,28),(17,41),(18,54),(20,24),(21,37),(22,50),(25,33),(26,46),(30,42),(31,55),(34,38),(35,51),(39,47),(44,56),(48,52),(57,77),(58,90),(59,103),(61,73),(62,86),(63,99),(64,112),(65,69),(66,82),(67,95),(68,108),(70,78),(71,91),(72,104),(75,87),(76,100),(79,83),(80,96),(81,109),(84,92),(85,105),(89,101),(93,97),(94,110),(98,106),(107,111),(113,129),(114,142),(115,155),(116,168),(117,125),(118,138),(119,151),(120,164),(122,134),(123,147),(124,160),(126,130),(127,143),(128,156),(131,139),(132,152),(133,165),(136,148),(137,161),(140,144),(141,157),(145,153),(146,166),(150,162),(154,158),(159,167),(169,177),(170,190),(171,203),(172,216),(174,186),(175,199),(176,212),(178,182),(179,195),(180,208),(181,221),(183,191),(184,204),(185,217),(188,200),(189,213),(192,196),(193,209),(194,222),(197,205),(198,218),(202,214),(206,210),(207,223),(211,219),(220,224)], [(1,201,60,149),(2,216,61,164),(3,175,62,123),(4,190,63,138),(5,205,64,153),(6,220,65,168),(7,179,66,127),(8,194,67,142),(9,209,68,157),(10,224,69,116),(11,183,70,131),(12,198,71,146),(13,213,72,161),(14,172,73,120),(15,187,74,135),(16,202,75,150),(17,217,76,165),(18,176,77,124),(19,191,78,139),(20,206,79,154),(21,221,80,113),(22,180,81,128),(23,195,82,143),(24,210,83,158),(25,169,84,117),(26,184,85,132),(27,199,86,147),(28,214,87,162),(29,173,88,121),(30,188,89,136),(31,203,90,151),(32,218,91,166),(33,177,92,125),(34,192,93,140),(35,207,94,155),(36,222,95,114),(37,181,96,129),(38,196,97,144),(39,211,98,159),(40,170,99,118),(41,185,100,133),(42,200,101,148),(43,215,102,163),(44,174,103,122),(45,189,104,137),(46,204,105,152),(47,219,106,167),(48,178,107,126),(49,193,108,141),(50,208,109,156),(51,223,110,115),(52,182,111,130),(53,197,112,145),(54,212,57,160),(55,171,58,119),(56,186,59,134)]])

64 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 7A 7B 7C 8A 8B 8C 8D 14A ··· 14I 28A ··· 28F 28G ··· 28R 56A ··· 56L order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 7 7 7 8 8 8 8 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 1 1 14 14 2 2 4 4 4 4 14 14 28 28 28 28 2 2 2 4 4 28 28 2 ··· 2 4 ··· 4 8 ··· 8 4 ··· 4

64 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + - + + + + + + - - + image C1 C2 C2 C2 C2 C2 C2 C2 C4 Q8 D4 D4 D7 D14 D14 C4×D7 C8⋊C22 C8.C22 Q8×D7 D4×D7 D8⋊D7 Q16⋊D7 kernel C56⋊(C2×C4) C28.Q8 C4.Dic14 C8⋊Dic7 C7×C2.D8 D7×C4⋊C4 C4⋊C4⋊7D7 C2×C8⋊D7 C8⋊D7 C4×D7 C2×Dic7 C22×D7 C2.D8 C4⋊C4 C2×C8 C8 C14 C14 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 8 2 1 1 3 6 3 12 1 1 3 3 6 6

Matrix representation of C56⋊(C2×C4) in GL6(𝔽113)

 4 67 0 0 0 0 106 109 0 0 0 0 0 0 37 15 37 15 0 0 4 52 4 52 0 0 76 98 37 15 0 0 109 61 4 52
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 104 0 0 0 0 25 0 0 0 0 0 0 0 0 104 0 0 0 0 25 0
,
 35 36 0 0 0 0 35 78 0 0 0 0 0 0 50 0 103 0 0 0 0 50 0 103 0 0 103 0 63 0 0 0 0 103 0 63

G:=sub<GL(6,GF(113))| [4,106,0,0,0,0,67,109,0,0,0,0,0,0,37,4,76,109,0,0,15,52,98,61,0,0,37,4,37,4,0,0,15,52,15,52],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,25,0,0,0,0,104,0,0,0,0,0,0,0,0,25,0,0,0,0,104,0],[35,35,0,0,0,0,36,78,0,0,0,0,0,0,50,0,103,0,0,0,0,50,0,103,0,0,103,0,63,0,0,0,0,103,0,63] >;

C56⋊(C2×C4) in GAP, Magma, Sage, TeX

C_{56}\rtimes (C_2\times C_4)
% in TeX

G:=Group("C56:(C2xC4)");
// GroupNames label

G:=SmallGroup(448,415);
// by ID

G=gap.SmallGroup(448,415);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,120,219,58,438,102,18822]);
// Polycyclic

G:=Group<a,b,c|a^56=b^2=c^4=1,b*a*b=a^13,c*a*c^-1=a^15,b*c=c*b>;
// generators/relations

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